↑

Properties

Label 20T254
Order \(2560\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $254$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,15,7,18,4,9,13,6,20)(2,11,16,8,17,3,10,14,5,19), (1,8,11,18,14)(2,7,12,17,13)(3,5,10,19,15)(4,6,9,20,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$
80:  $C_2^4 : C_5$
160:  $C_2 \times (C_2^4 : C_5)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2 \times (C_2^4 : C_5)$

Low degree siblings

20T251 x 8, 20T254 x 7, 40T1880 x 8, 40T1882 x 8, 40T1958 x 2, 40T1959 x 2, 40T1972 x 4, 40T1973 x 4, 40T2011 x 2, 40T2013 x 2, 40T2021 x 4, 40T2114 x 4, 40T2123 x 8, 40T2124 x 8, 40T2125 x 8, 40T2129 x 8, 40T2130 x 8, 40T2131 x 8, 40T2136 x 2, 40T2138 x 2, 40T2142 x 2, 40T2143 x 2, 40T2147 x 4, 40T2165 x 4, 40T2168 x 4, 40T2173 x 8, 40T2174 x 8, 40T2205 x 4, 40T2208 x 4, 40T2243 x 4, 40T2245 x 8

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 9,11)(10,12)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $2$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,11)(10,12)(13,14)(19,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 4, 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,14)(15,16)(17,20,18,19)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 7, 8)(13,15,14,16)(19,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 1, 4, 2, 3)( 7, 8)( 9,10)(11,12)(13,15,14,16)(17,18)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,15,18, 9, 6)( 2,16,17,10, 5)( 3,14,19,11, 8)( 4,13,20,12, 7)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,18, 6,15, 9)( 2,17, 5,16,10)( 3,19, 8,14,11)( 4,20, 7,13,12)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 6, 9,18,15)( 2, 5,10,17,16)( 3, 8,11,19,14)( 4, 7,12,20,13)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 9,15, 6,18)( 2,10,16, 5,17)( 3,11,14, 8,19)( 4,12,13, 7,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $(11,12)(15,16)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 1, 2)( 3, 4)(11,12)(15,16)(17,20)(18,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 5, 6)( 7, 8)(11,12)(13,14)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $20$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,20)(18,19)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $4$ $( 1, 4)( 2, 3)( 9,11,10,12)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $20$ $4$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,11,10,12)(13,14)(15,16)(17,20,18,19)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $20$ $4$ $( 1, 4)( 2, 3)( 9,11,10,12)(13,14)(15,16)(17,19,18,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $20$ $4$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,11,10,12)(17,19,18,20)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $80$ $4$ $( 1, 4, 2, 3)( 5, 7)( 6, 8)( 9,11,10,12)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)$
$ 10, 10 $ $256$ $10$ $( 1,15,17,11, 7, 4,13,20, 9, 6)( 2,16,18,12, 8, 3,14,19,10, 5)$
$ 10, 10 $ $256$ $10$ $( 1,18, 7,13,12, 3,19, 5,16, 9)( 2,17, 8,14,11, 4,20, 6,15,10)$
$ 10, 10 $ $256$ $10$ $( 1, 6, 9,18,13, 4, 7,12,19,16)( 2, 5,10,17,14, 3, 8,11,20,15)$
$ 10, 10 $ $256$ $10$ $( 1, 9,15, 5,17, 3,11,13, 7,20)( 2,10,16, 6,18, 4,12,14, 8,19)$

Group invariants

Order:  $2560=2^{9} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.